Problem 71 - Precalculus : Building Concepts and Connections

Question

The concentration \(C(t)\) of a drug in a patient's bloodstream \(t\) hours after administration is given by $$ C(t)=\frac{10 t}{1+t^{2}} $$ where \(C(t)\) is in milligrams per liter. (a) What is the drug concentration in the patient's bloodstream 8 hours after administration? (b) Find the horizontal asymptote of \(C(t)\) and explain its significance.

Step-by-Step Solution

Answer

Part (a): The drug concentration in the patient's bloodstream 8 hours after administration is approximately 1.23 milligrams per liter. Part (b): The function \(C(t)\) has a horizontal asymptote at \(C(t) = 10\), meaning that the concentration of the drug approaches 10 mg/L as time goes to infinity. This maximum concentration is the therapeutic limit for this drug in the bloodstream.

Step 1 - Calculation of drug concentration

Substitute \(t = 8\) into the given function \(C(t)=\frac{10 t}{1+t^{2}}\) to find the concentration 8 hours after administration. That gives \(C(8) = \frac{10 * 8}{1 + 8^{2}} = \frac{80}{1 + 64} = \frac{80}{65}\) or approximately 1.23 milligrams per liter.

Step 2 - Identification of horizontal asymptote

A rational function \(f(x) = \frac{p(x)}{q(x)}\) has a horizontal asymptote as \(x \rightarrow \infty\) if the degree of \(p(x)\) and \(q(x)\) are equal. It is given by the ratio of the coefficients of the highest degree terms. Since the degree of the numerator and the denominator are equal in \(C(t) = \frac{10t}{1 + t^{2}}\), it has a horizontal asymptote, which can be calculated by taking limit as \(t \rightarrow \infty\). This gives \(C_{asymptote} = \lim_{{t \to \infty }} \frac{10t}{1 + t^{2}} = \lim_{{t \to \infty }} \frac{10}{\frac{1}{t} + t} = 10\). The graphs of such a function approach this line but never reach it.

Step 3 - Interpreting the horizontal asymptote

The horizontal asymptote at \(C(t) = 10\) indicates that the concentration of the drug in the bloodstream approaches 10 milligrams per liter as time goes to infinity. This means the concentration will get closer and closer to 10 mg/L over time but never reach or surpass it. This is the maximum concentration the drug can reach in the bloodstream under continuous administration.

Key Concepts

Rational Functions

Rational functions are ratios of polynomial functions, where one polynomial is divided by another.
The general form of a rational function is \[\begin{equation}R(x) = \frac{p(x)}{q(x)}\end{equation}\]where both p(x) and q(x) are polynomials and q(x) is not the zero polynomial. In the exercise, the concentration function \[\begin{equation}C(t)=\frac{10 t}{1+t^{2}}\end{equation}\]is a rational function of time \[\begin{equation}t\end{equation}\]since both the numerator \[\begin{equation}10t\end{equation}\]and the denominator \[\begin{equation}1+t^{2}\end{equation}\]are polynomials in terms of t.
Rational functions often describe real-world situations where there is a rate of change that is contingent upon the current value of the variable, such as the concentration of the drug in the bloodstream over time in our exercise. They can show growth or decline and can have varying rates throughout their domain.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a function approaches as the independent variable, usually x or in our case t, tends towards infinity or negative infinity.
In the context of rational functions, we determine the horizontal asymptote by comparing the degrees of the polynomials in the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0.
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

For our drug concentration function \[\begin{equation}C(t)=\frac{10 t}{1+t^{2}}\end{equation}\]the numerator and denominator both have degrees of 1, hence the function will have a horizontal asymptote. This line, which we find not by intersection but by limits, represents a boundary that the output value of the function will never exceed, indicating the maximum possible concentration of the drug in the bloodstream in this case.

Limits in Calculus

Limits are fundamental in calculus and help describe the behavior of functions as inputs approach a certain value. The limit of a function f(x) as x approaches a value a essentially answers the question, 'What value does f(x) get closer to as x gets arbitrarily close to a?' When solving for horizontal asymptotes mathematically, we make use of limits to infinity.
The notation \[\begin{equation}\lim_{{x \to a}} f(x)\end{equation}\]is used to represent the limit of f(x) as x approaches a. In the example of drug concentration in a bloodstream, to find the asymptotic concentration as time goes to infinity, we calculate it using the limit \[\begin{equation}\lim_{{t \to \infty}} \frac{10t}{1 + t^{2}} = 10\end{equation}\]This helps us understand how the drug will behave over a long period and is essential for determining long-term dosing and therapeutic levels in medical applications.